june 2009 markscheme

Mark Scheme Summer 2009

IGCSE

IGCSE Mathematics (4400)

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4400 IGCSE Mathematics Summer 2009 3

Contents 1. Paper 1F Mark Scheme 5

2. Paper 2F Mark Scheme 15

3. Paper 3H Mark Scheme 21

4. Paper 4H Mark Scheme 35


4400 Paper 1F Mark Scheme

Except for questions* where the mark scheme states otherwise, the correct answer, unless clearly obtained by an incorrect method, should be taken to imply a correct method. [* Questions 15(b) and 18(b)]

Q Working Answer Mark Notes 1 a 6012 1 B1 cao b 6800 1 B1 cao c tens 1 B1 Accept 80, 10, T d 803 1 B1 cao Total 4 marks 2 a 54 63 2 B2 B1 each b eg Add 9, multiples of 9, 9 times table 1 B1 c 180 1 B1 cao Total 4 marks 3 ai 9 40 pm 2 B1 Allow 20 to 10 pm ii 21 40 B1 cao b −2 1 B1 cao c −8 indicated 1 B1 Allow + ½ division Total 4 marks


4 a 75 1 B1 cao b USA 1 B1 Accept any clear indication c bar 1 B1 Accept 25 < bar < 30 Total 3 marks 5 ai line 2 B1 ii isosceles B1 b lines 2 B2 for 4 correct

B1 for 2 correct ci octagon 2 B1 ii eg angles not all equal B1 di 0 2 B1 cao ii 2 B1 cao ei

52 2 B1 cao

ii 0.4 B1 ft from (i) if denominator is 3 or 5 but not if denominator is 2 or 4

If (i) is ""32 (ii) must be 6.0 & oe or have at least 2

decimal places rounded or truncated Total 10 marks


6 ai 22 24 4 B1 cao ii 28 B1 cao iii 25 B1 cao iv 23 or 29 B1 bi

91 3 B1

ii 95 M1

A1 denominator 9 numerator 5

Total 7 marks 7 ai 2.645751311 2 B1 for at least 5 figures ii 2.65 B1 ft from “2.645…” if at least 3 dp bi 0.0841 2 B1 cao ii 0.08 B1 ft from “0.0841” if of equal difficulty c 3.375 + 0.4 2 M1 for 3.375 or 0.4 3.775 A1 cao Total 6 marks 8 a 1 4 4 5 6 10 10 10 10 10

or 2

110 + or 215 or 6, 10

2 M1 for a clear attempt to list in order

8 A1 cao b 9 2 B2 B1 for 1-10, 10 − 1 Total 4 marks


9 a 4q 1 B1 Accept 4×q, q4 etc b 5np 1 B1 Do not accept × signs

Accept n5p, 5pn, 5(pn) etc c 7 1 B1 cao d 8y = 5 + 1 or 8y = 6 2 M1 May be implied by correct answer ¾ oe A1 Total 5 marks 10 a eg 0.666…, 0.7, 0.65, 0.625 2 B2

107

32

2013

85

for 107

32

2013

85 or for correct decimal equivalents

B1 for 3 fractions in correct order or for 2 fractions correctly converted to decimals (at least 2 dp rounded or truncated) or for 2 fractions expressed as equivalent fractions with a denominator of 120

b 125

129 − 2 M1 Accept

2410

2418 − or

4820

4836 −

124 A1 Accept

248 or

4816

Total 4 marks 11 a

248180 −

2 M1

66 A1 cao b 180 − “66” or 114 or

∠ ABC = “66°” 3 M1

360 − (69 + 106 + “114”) or 360 − (106 + 69 + 48 + “66”)

M1

71 A1 ft from “66” Total 5 marks


12 a

52

80 × , 580

2× 2 M1 Also award for 80 : 32 or 32 : 80

32 A1 cao b 3 + 1 or 4 2 M1 Also award for 60 : 20 or 20 : 60 20 A1 cao Total 4 marks 13

248180 −

3 M2

for 40 × 13.25 oe or 7956040

× oe

M1 for )156013(6040

+××

or for 40 × time eg 40 × 13.15 or 526 seen or 40 × 795 or 40 × 13. …

530 A1 cao Total 3 marks 14 correct enlargement

vertices (10,10) (15,10) (15,20) 3 B3 B2 for translation of correct shape

or 2 vertices correct or for enlargement 1½, centre (0, 0) B1 for one side correct length Allow ½ square tolerance for both vertices and lengths of sides of triangle

Total 3 marks


15 a 2×(12×7+7×5+12×5)

or 2 × (84 + 35 + 60) 2 M1 for correct substn or 179 seen

358 A1 for correct substn or 179 seen for correctly collecting Ls or constants or both M1 for correct substitution in given formula or in a correct rearrangement of the given formula in which L is not the subject

b 12L+16 = 70 or 8L + 4L = 54 or 12L = 54

6L + 8 = 35 or 4L + 2L = 27 or 6L = 27

3 M2

eg 70=2(4L + 2×4 + 2L) or 70=2(4L + 8 + 2L) or 35=4L+ 2×4 + 2L or 35=4L+ 8 + 2L or 70 − 2×2×4 = 8L + 4L or 35 − 2×4 = 4L + 2L

4.5 oe A1 depends on M2 Total 5 marks 16 a

85010014

× 2 M1

119 A1 cao b

760266

or 0.35 2 M1

35 A1 cao c

3.0204

or 30204

or 6.8 or 3

204 or 68

2 M1

680 A1 cao Total 6 marks


17 Examples of complete, correct explanations

(i) 10 × 0.35 or 3.5 seen (may be in 10

5.3 ) AND

can’t have half beads or there must be a whole number of (red) beads

(ii) 213 red beads is impossible

(iii) 207 AND there are (only)10 beads

or you need 20 beads (iv) The probability of any bead/a red bead must be tenths or must have 1 decimal place or must have 1 significant figure (v) Gives at least two examples that the

probability of taking a red bead is 10n where

2 < n < 9 e.g. states 0.3 and 0.4

2 B2 for a complete, correct explanation B1 for a partially correct explanation Examples of partially correct explanations

(i) 101 or 0.1 seen

(ii) Gives one example that the probability of

taking a red bead is 10n where 2 < n < 9

(iii) There would be 3.5 red beads. (iv) 10 × 0.35 = 3.5

(v) 0.35 = 207

Treat statements like ‘Don’t know the number of red beads’ as irrelevant.

Total 2 marks


18 a p(p + 7) 2 B2 Also accept (p + 0)(p + 7) for B2

B1 for factors which, when expanded and simplified, give two terms, one of which is correct. SC B1 for p(p + 7p)

b 5x = 2 or −5x = −2 3 M2 for 5x = 2 or −5x = −2 or

52

55

=x

M1 for 4 = 5x + 2 or 5x = 4 − 2 or −5x = 2 − 4 or 5x − 2 = 0

52 or 0.4 A1 dep on at least M1

c t9 1 B1 cao d 12y + 15 −10y − 15 2 M1 for 3 correct terms inc correct signs

or for 12y + 15 − (10y + 15) 2y A1 Accept 2y ± 0 Total 8 marks

M1 for finding at least three products f×x consistently within intervals (inc end points) and summing them

19 10×8 + 30×24 + 50×5 + 70×2 + 90 × 1 or 80 + 720 + 250 + 140 + 90 or 1280

4

M1 (dep) for use of halfway values

40"1280"

M1 (dep on 1st M1) for division by 40

or for division by their 8+24+5+2+1



20 ½ × 10 × 12 or 60 3 M1 for area of one triangle 13 × 15 + 13 × 15 + 10 × 15

or 195 + 195 + 150 or 540 M1 for

13 × 15 + 13 × 15 + 10 × 15 oe 660 A1 cao Total 3 marks 21 a 1 3 9 27 2 B2 −B1 for eeoo

or any repetition b Yes and gives an explanation which either

refers specifically to the members of A and their properties eg All the factors of 27 are odd. None of the factors of 27 are even. 2, 4, 6, 8 aren’t factors of 27. or gives a general explanation which shows understanding of the statement eg A and C have no members in common. The intersection of A and C is empty.

1 B1 for ‘Yes’ and an acceptable explanation Do not accept an explanation which merely lists, without comment, the members of both sets. Do not accept an explanation which includes the symbol ∩ with no indication of its meaning.

Total 3 marks 22 sin 3 M1 for sin

9.76.3

or 0.4556… A1

for 9.76.3

oe

or 0.4556…

or M1 for cos and

9.7"45.49"

following

correct Pythagoras and A1 for 0.8901… or M1 for tan and

"45.49"

6.3 following

correct Pythagoras and A1 for 0.5119…

Total 3 marks


4400 Paper 2F Mark Scheme

Q Working Answer Mark Notes 1 ai 998 1908 1990 1998 2001 1 B1 ii 2001 1 B1 iii 1908 1 B1 iv 1998 – 998 1 B1 B0 for 998-1998 bi 3478 1 B1 ii 8734 2 B2 B1 for 8374 Total 7 marks 2 ai kite 1 B1 Allow mis-spellings (any recognisable attempt) ii parallelogram 1 B1 Allow mis-spellings (any recognisable attempt) iii trapezium 1 B1 Allow mis-spellings (any recognisable attempt) bi acute 1 B1 Allow mis-spellings (any recognisable attempt) ii reflex 1 B1 Allow mis-spellings (any recognisable attempt) Total 5 marks 3

i ii iii

A at 0.5 + 2mm B at 1 ± 2mm

C > 0 & < 0.25

1 1 1

B1 B1 B1

If no Xs, mark point on line level with middle of letter A, B or C If no letters then no marks

Total 3 marks 4 a 5 x 4 + 12

32 2

M1 A1

cao

b (47-12) ÷ 5 7

2

M1 A1

M1 for 47-12 or 35 or 47÷5 or 9.4 or 5”n”+12=47 cao

Total 4 marks


5 a 1, 3, 11, 33 2 B2 B2 fully correct (no additions or errors)

B1 for any two correct factors 3 correct & 1 wrong = B1

b 46 1 B1 No embedded answers i.e. 462 =2116 c 243 1 B1 d 26 1 B1 No embedded answers i.e. 263 =17576 Total 5 marks 6 7 x 1.20 + 6 x 0.75 (= 12.9)

20 – “12.9”

7.1(0)

3

M1 M1 A1

condone omission of final zeros dep

Total 3 marks 7 a 6 1 B1 b Attempt to add all the numbers

“88” ÷ 8

11

3

M1 M1 A1

dep

If ans = 76.6(25) M2 A0 c 11 1 B1 ft (b) Total 5 marks 8 a 3 + 5 + 3 + 5 oe

16 2

M1 A1

b 46.8 ÷ 7.2 6.5

2

M1 A1

Total 4 marks


9 ai 9/36 1 B1 ii 4/20 1 B1 b 2/3 x 9/5

x/9 and y/9

6/9 ÷ 5/9

18/15 or 6/5

3

M2

M2 A1

M1 for inverting 2nd fraction i.e. 9/5 oe or M1 for 2 correct fractions with a common denominator of a multiple of 9 M1 correct numerators and intention to divide Any fraction equivalent to 1 1/5

Do not allow decimal conversions Total 5 marks 10 a 12

cm2 sq cms 3

B2 B1

B1 for 11 to 13 or 3 × 4 ind

b Correct + 2 mm 2

B2 B1 for any 2 vertices correct + 2 mm or correct size, shape & orientation

Total 5 marks 11 a (10 + 5) x 4

60 2

M1 A1

brackets necessary unless answer correct

b 28 ÷ 4 – 5 2

2

M1 A1

allow 23 ÷ 4 or 5.75 (i.e. reverse operations but wrong order)

c -8 ÷ 4 – 5 or -2 - 5 -7

2

M1 A1

allow -13 ÷ 4 or -3.25 (i.e. reverse operations but wrong order)

d (x + 5) × 4 or 4x + 20 oe 2 B2 B1 for x+5×4 or x+20 or 4 x +5 or “y=” 4x+5 B0 for x=4x+5

Total 8 marks


12 a 250 x 1.85

462.5(0)

2 M1 A1

462 or 463 = M1 A0

b 320 ÷ 1.85 172(.97…)

2

M1 A1

awrt 173

c 1 ÷ 1.85 oe 0.54

2

M1 A1

e.g “172.97” ÷ 320 or 250 ÷ “462.5” awrt 0.54

Total 6 marks 13 a 90 ÷ 40(=2.25) or 12 ÷ 40(=0.3) or 40 ÷ 12(=3 1/3)

then “2.25” x 12 or “0.3” x 90 or 90 ÷ “3 1/3” (scale factors) (students per degree) (degrees per student)

27

3

M1

M1 A1

or M2 for 12 x 90 ÷ 40 M1 for 9 x 12 (=108) then M1(dep) for “108” / 4 dep cao

b 130/240 x 360 195o

2

M1 A1

M1 for 130/240

cao Total 5 marks 14 a x - 5 1 B1 Accept y=x-5 not x=x-5 or 0=x-5 bi 3(x – 5) = 39 or 3x-15=39 or x-5=13 M2 M1 for 3x – 5 = 39 ii 3x = 54 or x - 5 =13

18 4

M1 A1ft

Allow full ft on ax +b =c from bi ans a>1 b,c ≠ 0 18 no wrong working = M1 A1

Total 5 marks 15 6 × (-9 + 1)

= -48 oe (-54+6)

-3

3

M1 M1 A1

allow without brackets M1 for -8 numerator correct (or or x -8) cao

Total 3 marks


16 67 ÷ 2 or (67 +1) ÷ 2 oe M1 attempt to find middle of frequencies of people 7 2 A1 cao look for mean (7.56..) rounded down M0 A0 Total 2 marks 17 a 2 x π x 40 oe

251 2

M1 A1

awrt 251

b 8 x 10 or 80 π x 32 (value rounding to 28.3 or 28.2) “8x10” – “π x 32”

51.7

4

M1 M1 M1 A1

Rectangle area Circle area dep on both M1’s awrt 51.7

Total 6 marks 18 a 1 – (0.3 + 0.1 + 0.4)

0.2oe 2

M1 A1

Look for answer in table Decimals, fractions, % only

b 0.3 + 0.4 0.7oe

2

M1 A1

Decimals, fractions, % only

Total 4 marks 19 a 5.12 + 3.22 (= 36.25)

√“36.25”

6.02

3

M1 M1 A1

M2 for 5.1/cos(tan-1-(3.2/5.1)) or 3.2/sin(tan-1-(3.2/5.1)) awrt 6.02

b tan selected (AB =) 6.5 x tan 32o

4.06

3

M1 M1 A1

sin 32o = AB/6.5/cos32 (AB =) sin 32o × 6.5/cos32 awrt 4.06

Total 6 marks


20 12–x=21 or 12-21=x or –x=21-12

[12 - 21 = x] or [-x = 21 – 12] oe

-9

3

M2

A1

[-x/3 = 7 – 12/3 ] or [12/3 - 7 = x/3 ] M1 for 12-x=3x7

Total 3 marks 21 A product of 3 or more factors of

which 2 are from 2,2,3,11

1,2,2,3,11or 2,2,3,11

2 x 2 x 3 x 11

3

M1

M2 A1

Product can be implied from a factor tree or repeated division These combinations can be implied from a factor tree or repeated division cao

Total 3 marks 22 [80/40] or [84/42]

√36 or 6

12

3

B1 B1 B1

Dep on both previous b1’s

(Accept 10 if 80/40, 6 used) Total 3 marks

Total 100 marks


4400 Paper 3H Mark Scheme

Except for questions* where the mark scheme states otherwise, the correct answer, unless clearly obtained by an incorrect method, should be taken to imply a correct method. [* Questions 5(b), 11(a), 13(a), 15(d), 20 and 21] Trial and improvement methods for solving equations score no marks, even if they lead to a correct solution.

Q Working Answer Mark Notes 1 a

52

80 × , 580

2× 2 M1 Also award for 80 : 32 or 32 : 80

32 A1 cao b 3 + 1 or 4 2 M1 Also award for 60 : 20 or 20 : 60 20 A1 cao Total 4 marks 2

40 × 13.25 or 7956040

× oe

3 M2 for 40 × 13.25 oe or 795

6040

× oe

M1 for )156013(6040

+××

or for 40 × time eg 40 × 13.15 or 526 seen or 40 × 795 or 40 × 13. …



3 correct enlargement

vertices (10,10) (15,10) (15,20) 3 B3 B2 for translation of correct shape

or 2 vertices correct or for enlargement 1½, centre (0, 0) B1 for one side correct length Allow ½ square tolerance for both vertices and lengths of sides of triangle

Total 3 marks 4 Examples of complete, correct explanations

(i) 10 × 0.35 or 3.5 seen (may be in 10

5.3 ) AND

can’t have half beads or there must be a whole number of (red) beads

(ii) 213 red beads is impossible

(iii) 207 AND there are (only)10 beads

or you need 20 beads (iv) The probability of any bead/a red bead must be tenths or must have 1 decimal place (v) Gives at least two examples that the

probability of taking a red bead is 10n where

2 < n < 9 e.g. states 0.3 and 0.4

2 B2 for a complete, correct explanation B1 for a partially correct explanation Examples of partially correct explanations

(i) 101 or 0.1 seen

(ii) Gives one example that the probability of

taking a red bead is 10n where 2 < n < 9

(iii) There would be 3.5 red beads. (iv) You can’t have half beads (v) 10 × 0.35 = 3.5

(vi) 0.35 = 207

Treat statements like ‘Don’t know the number of red beads’ as irrelevant.

Total 2 marks


5 a p(p + 7) 2 B2 Also accept (p + 0)(p + 7) for B2

B1 for factors which, when expanded and simplified, give two terms, one of which is correct. SC B1 for p(p + 7p)

b 5x = 2 or −5x = −2 3 M2 for 5x = 2 or −5x = −2 or

52

55

=x

M1 for 4 = 5x + 2 or 5x = 4 − 2 or −5x = 2 − 4 or 5x − 2 = 0

52 or 0.4 A1 for 4 correct

B1 for 2 correct c t9 1 B1 cao d 12y + 15 − 10y − 15 2 M1 for 3 correct terms inc correct signs

or for 12y + 15 − (10y + 15) 2y A1 Accept 2y + 0 Total 8 marks 6 a

760266

or 0.35 2 M1

35 A1 cao b

3.0204

or 30204

or 6.8 or 3

204 or 68

2 M1



7 sin 3 M1 for sin

9.76.3

or 0.4556… A1 for

9.76.3

oe

or 0.4556…

or M1 for cos and

9.7"45.49"

following correct

Pythagoras and A1 for 0.8901…

or M1 for tan and "45.49"

6.3

following correct Pythagoras and A1 for 0.5119…

27.1 A1 for answer rounding to 27.1 Total 3 marks 8 a 1 3 9 27 2 B2 −B1 for eeoo

or any repetition b Yes and gives an explanation which either refers

specifically to the members of A and their properties eg All the factors of 27 are odd. None of the factors of 27 are even. 2, 4, 6, 8 aren’t factors of 27. or gives a general explanation which shows understanding of the statement eg A and C have no members in common. The intersection of A and C is empty.

1 B1 for ‘Yes’ and an acceptable explanation Do not accept an explanation which merely lists, without comment, the members of both sets. Do not accept an explanation which includes the symbol ∩ with no indication of its meaning.

c

2 B2 B1 for B ⊂ A B1 for A ∩C = Ø and B ∩ C = Ø Ignore any individual members shown on the diagram. Mark the layout which must be labelled

Total 5 marks

E

A

B C


9 22 9.57.4 +

= 22.09 + 34.81 = 56.9

4 M1 for squaring & adding

22 9.57.4 + M1 (dep) for square root

7.5432… A1 for value which rounds to 7.54 2.84 A1 for answer which rounds to 2.84

(2.84320…) Total 4 marks

M1 for finding at least three products f×x consistently within intervals (inc end points) and summing them

10 a 10×8 + 30×24 + 50×5 + 70×2 + 90 × 1 or 80 + 720 + 250 + 140 + 90 or 1280

4

M1 (dep) for use of halfway values

40"1280"

M1 (dep on 1st M1) for division by 40

or division by their 8+24+5+2+1

32 A1 cao b d = 25 indicated on graph 2 M1 12 or13 A1 Accept 12 – 13 inc c 10 and 30 or

4110 and

4330

indicated on cumulative frequency axis or stated

2 M1

14 - 17 inc A1 Total 8 marks


11 a 10x−15y=45

10x+8y=22 8x−12y=36 15x+12y=33

4 M1 for coefficients of x or y the same followed by correct operation or for correct rearrangement of one equation followed by substitution in

the other eg 113

9245 =⎟

⎠⎞

⎜⎝⎛ −

+x

x

For both approaches, condone one arithmetical error

y = −1 x = 3 A1 cao dep on M1 M1 (dep on 1st M1) for substituting for other variable 3 −1 A1 cao dep on all preceding marks b 3, −1 1 B1 ft from (a) Total 5 marks 12 a 1.5 × 108 2 M1 for 1.5 × 10m A1 if m = 8 b 7.2 × 10−1 2 M1 for 7.2 × 10n or 0.72 oe with digits 72

eg 72 × 10−2 A1 if n = −1 Total 4 marks


for correctly collecting Ls or constants or both M1 for correct substitution in given formula or in a correct rearrangement of the given formula in which L is not the subject

13 a 12L+16 = 70 or 8L + 4L = 54 or 12L = 54

6L + 8 = 35 or 4L + 2L = 27 or 6L = 27

3 M2

eg 70=2(4L + 2×4 + 2L) or 70=2(4L + 8 + 2L) or 35=4L+ 2×4 + 2L or 35=4L+ 8 + 2L or 70 − 2×2×4 = 8L + 4L or 35 − 2×4 = 4L + 2L

4.5 oe A1 depends on M2 a alternative method

)(22

HWHWA

L+

−= oe

3 M1 for making L the subject of the given formula

eg

)24(242270

+××−

M1 for correct substitution into a correct

expression for L

4.5 oe A1 depends on both method marks


13 b A=2LW+2WH+2HL

or HLWHLWA

++=2

4 M1 for a correct equation following expansion or division by 2 May be implied by second M1

A−2HL=2LW+2WH

or WHLWHLA

+=−2

M1 for correct equation with W terms isolated

A−2HL=2W(L+H) or A−2HL=W(2L+2H)

or )(2

HLWHLA

+=−

M1 for correct equation with W as a factor

)( HLHLA

+−

22 or

HLHLA22

2+−

or HL

HLA

+

−2 oe

A1

Total 7 marks 14 ai 47 2 B1 cao ii alternate angles B1 Award this mark if ‘alternate’ appears b 124 1 B1 cao ci 47 2 B1 cao ii angle between a chord and a tangent

= angle in the alternate segment B1 Accept ‘alternate segment’

Total 5 marks


15 a 12 1 B1 cao Do not accept (3, 12) b 0.2 3.6 6.1 or 6.2 or values rounding to these 2 B2 for all 3 correct solutions

(B1 for 2 correct solutions or for 3 coordinates with correct solutions as x-coordinates)

c 5 seen 2 M1 0 A1 cao d tan drawn at (1, 16) 3 M1 tan or tan produced passes between points

(0.5, 11 < y < 13) and (1.5, 19 < y < 21)

ediffrerenc horizontaldifference vertical

M1

finds their difference horizontal

difference vertical

for two points on tan or finds the intercept of their tangent on the y-axis and substitutes y = 16, x = 1 and their c into y = mx + c

or finds their difference horizontal

difference vertical

for two points on curve, where one of the points has an x-coordinate between 0.5 and 1 inc and the other point has an x-coordinate between 1 and 1.5 inc

6–10 inc A1 dep on both M marks Total 8 marks


16 a π × 42 + π × 4 × 9 2 M1 163 A1 for ans rounding to 163

(π→ 163.3628… 3.14→ 163.28 3.142→ 163.384)

b

46

or 1.5 oe or 6 : 4 oe

or 64

oe or 4 : 6 oe

2 M1 May be implied by 13.5 or 12.09… Also award for cube of any correct values or cube of correct ratios

3.375 oe A1 for 3.375 or 833 or

827 oe

Accept 3.38 if M1 scored Do not award A1 if slant heights used as

h in hrV 231π=

Total 4 marks


17 i

42

53 × 5 M1

206 or

103 A1

Sample space method – award 2 marks for a correct answer, otherwise no marks

ii 41

51 × × 2 + “

206 ”

or 41

52 × + “

206 ”

M1 for 41

51 ×

or 41

52 ×

M1 for

complete sum

Award M0 M0 A0 for 52

51

51 =+

Sample space method – award 3 marks for a correct answer, otherwise no marks

SC

M1 for 51

51 ×

or 251

208 or

52 oe

A1

M1 for

their(i)251

51 +××

Sample space method - award 2 marks for

2511 otherwise no marks

Total 5 marks

for factorising numerator as )3)(15( +− xx 18

)125(2

)3)(15(2 −

+−

x

xx

)15)(15(2)3)(15(

−++−xx

xx

4 B1

B1

B1

for factorising denominator

as )125(2 2 −x

for factorising 125 2 −x as )15)(15( −+ xx

or B2 for factorising denominator as

)210)(15( +− xx or

)210)(15( −+ xx

)15(23+

+x

xor

2103++x

x

B1

Total 4 marks


19 2 × 6 sin39°

or 2 × 6 cos51° or 62 + 62 − 2×6×6cos78°

or °°

51sin78sin6

6 M1

7.551… A1 for answer rounding to 7.55

eg 1236078

××π M1

for 36078

oe inc 0.2166… rounded or

truncated to at least 3 decimal places

or for 78360

oe inc 4.6153… rounded or

truncated to at least 3 decimal places M1 for 12×π or for 62 ×π

(π→ 37.699… 3.14→ 37.68 3.142→ 37.704)

8.16 - 8.17 inc oe inc 5

13π, 2.6π oe

A1 for 8.17 or better (π→ 8.168… 3.14→ 8.164 3.142→ 8.1692)

15.7 A1 for ans rounding to 15.7 (π→ 15.7199… 3.14→15.7158… 3.142→15.7202…)

Total 6 marks 20 225 seen 3 B1 225 or 15 B1 Award B1 for 15 only if 225 seen

60 B1 cao Award only if preceding 2 marks scored

Total 3 marks


21 (x + 4)2 = x2 +(x + 6)2 −2x(x + 6)cos60°

or cos 60°=)6(2

)4()6( 222

++−++

xxxxx

5 M1

x2 + 4x + 4x + 16 or x2 + 8x + 16 and x2 + 6x + 6x + 36 or x2 + 12x + 36

B1 dep on M1 for correct expansion of (x + 4)2 and (x + 6)2 in correct statement of Cosine Rule

x2 + 8x + 16 = x2 + x2 + 12x + 36 − x2 − 6x or x2 + 6x = x2 +12x + 36 + x2 − x2 − 8x − 16 oe

B1 for correctly dealing with cos 60° and obtaining a correct equation with no fractions and no brackets

Omitted brackets may be implied by correct subsequent working.

2x = 20 oe B1 for correct linear equation e.g. 2x = 20 −2x = −20, 4x = 40, 2x − 20 = 0

10 A1 cao dep on all preceding marks Total 10 marks


4400 Paper 4H Mark Scheme

Except for questions 9, 11, 21 (where the marking scheme states otherwise), unless clearly obtained by an incorrect method, a correct answer should be taken to imply a correct method. Trial and improvement methods for solving equations score no marks, even if they lead to correct answers.

Q Working Answer Mark Notes 1 2/3 x 9/5

6a/9a and 5a/9a

6a/9a ÷ 5a/9a

18/15 or 6/5

3

M2

M2

A1

M1 for inverting 2nd fraction i.e. 9/5 or M1 2 correct fractions with common denominators of a multiple of 9 correct numerators and intention to divide any fraction equivalent to 11/5

Do not allow decimal conversions Total 3 marks

2 i 3x -15 = 39 or 3(x – 5) = 39 or x-5=39/3 B3

do not accept x-5 =13 B2 for 3x – 5 = 39 if x-5 seen otherwise B1 B1 for x-5 seen B0 for x= 39/3 +5 oe

ii 3x = 54 or x - 5 = 13

18

5

M1 A1

ft from any linear equation ax+b=c a>1 b,c ≠ 0 ax= c-b or x=c/a – b/a

18 with no working for answer in i) or ii) gets M1 A1 Total 5 marks


3 6 × (-9 + 1) or -8 seen M1 allow 6 x -9 + 1 -48 or -54+6 M1 Accept )2(6 − or )83( x -8

-3 3 A1 Total 3 marks

4 67 ÷ 2 or (67 +1) ÷ 2 oe

7

2

M1

A1

attempt to find middle of cumulative frequency or listing of people. cao look for mean (7.56..) rounded down (M0 A0)

Total 2 marks

5 a 2 x π x 40 oe 251

2

M1 A1

answer rounding to 251

b 8 x 10 or 80 π x 32 (awrt 28.2 or 28.3) “8x10” – “π x 32”

51.7

4

M1 M1 M1 A1

dep on both M1’s answer rounding to 51.7

Total 6 marks

6 a 1 – (0.3 + 0.1 + 0.4) 0.2oe

2

M1 A1

Look for answer in table if missing from answer line

b 0.3 + 0.4 0.7oe

2

M1 A1

Total 4 marks


7 a Correct + 2 mm

2 B2 B1 for any 2 vertices correct + 2 mm

or translation of correct image b

Translation

⎟⎟⎠

⎞⎜⎜⎝

⎛−54

2

B1

B1

translate or translated or -4 in x dir’n, or 4 to left or 4 west (not backwards or across) AND 5 in y dir’n or 5 up or 5 north (not (-4,5) or vectors without brackets)

penalise contradictions Total 4 marks 8 a 5.12 + 3.22 (= 36.25)

√“36.25”

6.02

3

M1 M1 A1

M2 for 5.1/cos(tan-1-(3.2/5.1)) or 3.2/sin(tan-1-(3.2/5.1)) Must be complete methods answer rounding to 6.02

b tan selected 6.5 x tan 32o

4.06

3

M1 M1 A1

sin 32o = “AB”/6.5/cos32 or “AB”/sin32 = 6.5/sin 58 (AB =) sin 32o × 6.5/cos32 or (AB=) sin 32 x 6.5 / sin 58 answer rounding to 4.06

Total 6 marks 9 12 – x = 21 or 12-21=x or-x=21-

12

-9

3

M2

A1

or [-x/3 = 7 – 12/3 ] or [12/3 - 7 = x/3 ] M1 for 12–x=3x7

(Answer only gains no marks) Total 3 marks 10 A product of 3 or more factors

of which 2 are from 2,2,3,11 1,2,2,3,11 or 2,2,3,11

2 x 2 x 3 x 11

3

M2

A1

M1 can be implied from a factor tree or repeated division M2 can be implied from a factor tree or repeated division product must be stated (not dots for product)

Total 3 marks


11 [80/40] or [84/42]

√36 or 6

12

3

B1 B1 B1

dep on both previous B1’s (Accept 10 only if 80/40, 6 used)

(Answer only gains no marks) Total 3 marks 12 a v/h in a correct ∆

½ oe 2

M1 A1

M1 A0 for ½ x

b y = “½”x + 2 oe 2 B2 B1 for “½”x + 2 or L= “½”x + 2 c y = “½”x + c 1 B1 c any number≠ 2 or letter or y = “0.5”x

or a line parallel to their b) Total 5 marks 13 a 60 1 B1 b y/7.5 = 4/5 oe

6 2

M1 A1

correct ratios or correct use of sf (0.8 or 1.25 or 1.5 or 2/3)

c [ z/5 = 3/4] oe or [z/7.5 = 3/”6”] 3.75

2

M1 A1

allow ft on their “6” or correct use of sf (0.8 or 1.25 etc) cao

Total 5 marks 14 a 1/4

binary tree structure all probs & labels correct

3

B1 B1 B1

P(tail) on Ist throw

b “1/4” x “1/4” 1/16 or 0.0625

2

M1 A1

ft their 2 tail branches cao

Total 5 marks


15 a 3c7d5 2 B2 B1 for c7 or d5 Accept 3 x c7 xd5

b 16x12y4 2 B2 B1 for 16 or x12 or y4 Accept 16 x x12 x y4

c 2(x – 3)/x(x – 3) 2/x

2

M1 A1

either factorisation correct. Accept (x±0) (2±0) Accept 2±0/x±0 Look for incorrect algebra

Total 6 marks 16 a (2x – 3)(x + 1) 2 B2 B1 for one correct factor or (2x + 3)(x - 1) (integers only) b “1.5” and “-1” 1 B1 both reqd ft (a) if 2 linear factors Total 3 marks 17

a 2x + 3 2 B2 B1 each term (accept 3x0)

b “-5” 1 B1 ft their ax + b (a, b ≠ 0) c “2x + 3” = 0

x = -3/2

(-3/2, -9/4) oe

3

M1 A1 A1

only ft their dy/dx, if ax + b (a, b ≠ 0) cao dependent on 2x+3=0 cao Answer dependent on 2x +3 =0 seen

Total 6 marks 18

a -x oe 1 B1 can be unsimplified

b x + y oe 1 B1 can be unsimplified c Unsimplified expression in terms of x

and y for PA or AP (either correct or ft from b) e.g.(AP=) “x+y”+y-½x or (PA=) ½x-y-“x-y”

-0.5x-2y

3

B2

B1

B1 Correct vector statement with at least 3 terms including AP or PA e.g.PA = PC + CA or AP = AC + CP can include x and/or y cao

Total 5 marks


19 a 80/150 x 15 or 4 x 2

(small squares) (freq den)

8

2

M1

A1

M1 for any fd value in correct position and no errors or 1 large square=2.5 leaves or 1 small square=1/10 (leaf) oe

b Freq 4-5 = 12 and ( freq 5-6 = 6 or freq 5-9=24) ½ ×(freq 4-5 + freq 5-6) or (½ x freq 4-5 + 1/8 x freq 5-9)

9

3

M1

M1

A1

12 & 6 seen or 12 & 24 or 60 & 30 (small squares) dep e.g. (0.5 x 12) +( 0.5 x 6) or (0.5 x12)+(1/8 x 24) or 1/10 x 90

Total 5 marks 20 ai

BM = 1 or CM =1 B1 (can be marked on diagram) allow cosine rule method

ii (AM2 =) 22 – 12 (= 3) (AM =) √(22 – 12) (= √3)

√3/2 or √¾

4

M1 M1 A1

(dependent on 1 line of Pythagoras or sine rule)

b (√3/2)2 + (1/2)2 = ¾ + ¼ oe

2

M1 A1

(√3/2)2 Must be seen allow 0.75 + 0.25 if M1 gained

Total 6 marks


21 a

22)1(2433 2

×−××−±−

4173 ±−

0.281 and -1.78

3

M1

M1

A1

allow one sign error both answers rounding to 0.281 & -1.78

(answer only gains no marks) b ( )

( ) 11

12=

+−+

xxxx

2(x+1)-x = x(x+1) x2 -2=0 oe

±√2 or ±1.41…

4

M1

M1

M1

A1

11)1(2

+=−+

xx

x or x

xx

=+

−1

2

removal of denominator correct gathering of terms answer rounding to ±1.41

(answer only gains no marks) Total 7 marks 22

a x x 105 + 0.1y x 105 = z x 105

x + 0.1y oe

2

M1 A1

M1 for 0.1y or (10x x 104 + y x 104=10z x 104) or (10x +y =10z)

bi

7.5

1

B1

ii

0.75 x 10n-m (= a x 10p)

n – m -1

2

M1 A1

0.75 and n-m seen (even in part i) )

Total 5 marks

Total 100 marks

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june 2009 markscheme

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